The Yaglom limit for branching Brownian motion with absorption and slightly subcritical drift

Consider branching Brownian motion with absorption in which particles move independently as one-dimensional Brownian motions with drift $-\rho$, each particle splits into two particles at rate one, and particles are killed when they reach the origin. Kesten (1978) showed that this process dies out with probability one if and only if $\rho \geq \sqrt{2}$. We show that in the subcritical case when $\rho>\sqrt{2}$, the law of the process conditioned on survival until time $t$ converges as $t \rightarrow \infty$ to a quasi-stationary distribution, which we call the Yaglom limit. We give a construction of this quasi-stationary distribution. We also study the asymptotic behavior as $\rho \downarrow \sqrt{2}$ of this quasi-stationary distribution. We show that the logarithm of the number of particles and the location of the highest particle are of order $\epsilon^{-1/3}$, and we obtain a limit result for the empirical distribution of the particle locations.